Spectral geometry of the Steklov problem (Survey article)

Abstract

The Steklov problem is an eigenvalue problem with the spectral parameter in the boundary conditions, which has various applications. Its spectrum coincides with that of the Dirichlet-to-Neumann operator. Over the past years, there has been a growing interest in the Steklov problem from the viewpoint of spectral geometry. While this problem shares some common properties with its more familiar Dirichlet and Neumann cousins, its eigenvalues and eigenfunctions have a number of distinctive geometric features, which makes the subject especially appealing. In this survey we discuss some recent advances and open questions, particularly in the study of spectral asymptotics, spectral invariants, eigenvalue estimates, and nodal geometry.

Stability in the Inverse Steklov Problem on Warped Product Riemannian Manifolds

“…In this paper, we are interested in studying the amount of information contained in the Steklov spectrum associated to the class of Riemannian manifolds (M, g). Recall [16] that the Steklov spectrum is defined as the spectrum of the Dirichlet-to-Neumann map Λ g (abbreviated later by DN map) associated to (M, g). More precisely, consider the Dirichlet problem −△ g u = 0, on M, u = ψ, on ∂M, (1.5) where ψ ∈ H 1 2 (∂M ).…”

“…We shall thus denote the Steklov eigenvalues (counted with multiplicity) by 0 = σ 0 < σ 1 ≤ σ 2 ≤ · · · ≤ σ k → ∞. (1.8) We refer the reader to the nice survey [16] and references therein. The main results of this paper are the following.…”

Nodal deficiency, spectral flow, and the Dirichlet-to-Neumann map

“…In particular, for a simple eigenvalue λ k , it was shown that δ(φ k ) = Mor (Λ + ( ) + Λ − ( )) (2) for sufficiently small > 0, where Λ ± ( ) denote the Dirichlet-to-Neumann maps for the perturbed operator ∆ + (λ k + ), evaluated on the positive and negative nodal domains Ω ± = {±φ k > 0}, and Mor denotes the Morse index, or number of negative eigenvalues. For more on the spectrum of Dirichlet-to-Neumann operators, see [8,11,1] and the recent survey [9]. Similarly, if φ * is an eigenfunction for a degenerate eigenvalue λ * , the same argument yields…”

Probability distribution of the boundary local time of reflected Brownian motion in Euclidean domains

“…Note that there is a family of operators M p parameterized by p ≥ 0. For a smooth enough boundary ∂Ω (here we skip conventional mathematical restrictions and rigorous formulation of the involved functional spaces, see [67][68][69][70][71][72][73][74] for details), M p is well-defined pseudo-differential self-adjoint operator. When the boundary is bounded, the spectrum of M p is discrete, i.e., there are infinitely many eigenpairs {µ…”